Name                                     MATH 116 PRACTICE MIDTERM 1 Please work out each of the given problems.  Credit will be based on the steps that you show towards the final answer.  Show your work.   PROBLEM 1  Find the derivative of the following functions A.    f(x)  =  ln(ln x)    Solution We use the chain rule.  We write         u  =  ln x        u'  =  1/x         f(u)  =  ln u    f '(u)  =  1/u We get                                 1        1                    1         f '(x)  =                           =                                                 x        ln x              x ln x   B.      Solution We write         and use the chain and product rules.  We have         u  =  ex ln x        u'  =  ex ln x + ex / x         f(u)  =  eu        f '(u)  =  eu so that           C.      Solution This problem is much easier if we use the properties of logarithms first.  We simplify to          now the derivative is simply         f '(x)  =  1   Find the following anti-derivatives. A.     Solution We first simplify the exponents using algebra to get         Now we use the power rule for integration to get         2/3 31/2 x3/2 + 1/2 ln|x| + 1/2 x8 + 2x1/2 + C   B.      Solution For this integral, we use u-substitution.  We let         u  =  4 - 3x5         du  =  -15x4 dx We get         C.      Solution We use u-substitution again         u  =  x + ex         du  =  1 + ex  We get           PROBLEM 3  During an experiment with a deadly new virus, Tom drops a flask of the virus on the floor.  Tom and his four lab aids are immediately infected.  During the next six hours they infect an additional forty unsuspecting people.  Assume the rate of spreading of the virus is proportional to the number of people who have been infected. A.  Write a differential equation that models this situation.  Be sure to label your variables.   Solution We let         t  =  the time in hours after the flask is dropped         P(t)  =  the number of people infected after t hours Then          dP                          =   kP        P(0)  =  5        P(6)  =  45          dt   B.   How long will it be until one million people have been infected?   Solution Since this is the standard exponential growth model, the solution to the differential equation is         P  =  P0 ekt Since P(0)  =  5, we have  P0  =  5  Hence         P  =  5 ekt Now use the fact that P(6)  =  45 to get         45  =  5 e6k         9  =  e6k         ln 9  =  6k                     ln 9         k  =                  =  0.3662                      6 Hence         P  =  5 e0.3662t  We want the time when P  =  1,000,000.  We write         1,000,000  =  5 e0.3662t          200,000  =  e0.3662t         ln 200,000  =  0.3662 t                     ln 200,000         t  =                              =  33.3                      0.3662 We can conclude that there will be 1,000,000 people infected in less than 34 hours.    A population of bacteria is growing at the rate of          dP              3000                 =                                     dt            1 + 0.25t where t is the time in days.  When t = 0, the population is 1000.  A. Write an equation that models the population P in terms of the time t.   Solution We need to solve the integral         We let          u  =  1 + 0.25t        du  =  0.25 dt We have         We have P(0)  =  1000.  Hence         1000  =  12,000 ln|1 + 0.25(0)| + C         1000  = 12,000 ln|1|  + C  =  12,000 (0) + C         C  =  1000 We can conclude that         P(t)  =  12,000 ln|1 + 0.25 t| + 1000 B. What is the population after 3 days?   Solution We plug in 3 for t to get         P(3)  =  12,000 ln|1 + (0.25)(3)| + 1000  =  7715 We can conclude that the population will be 7715 after 3 days. C. After how many days will the population be 12,000? Solution  We set P  =  12,000 and solve for t.         12,000  =  12,000 ln|1 + 0.25 t| + 1000         11,000  =  12,000 ln|1 + 0.25 t|         11/12  =  ln|1 + 0.25 t|         e11/12  =  1 + 0.25 t         e11/12 - 1  =  0.25 t         t  =  4(e11/12 - 1)  =  6 We can conclude that after 6 days, the population will be 12,000.   Extra Credit:  Write down one thing that your instructor can do to make the class better.